I'm interested in open problems in Galois theory.
It's not necessary for them to be well known or considered important, but they have to be mostly Galois-theoretic, that is, not number theory or geometry with some relation to Galois theory (there's a lot of those!).
Also, I'm well aware of the inverse Galois problem, so avoid that one on any of its forms.
Note. The question was closed as a duplicate of this other question, but even a superficial read reveals that they are exactly complementary. The accepted answer answers that question, but not this one. Please reconsider!
There are Abhyankar’s conjectures in Galois theory, namely Abhyankar’s Conjecture for affine curves, Abhyankar’s Inertia Conjecture and Abhyankar’s Affine Arithmetical Conjecture. All three conjectures are about Galois theory in positive characteristic, related to algebraic and etale fundamental groups, and Galois covers. The last two conjectures are still open. A survey is given here.